# Questions tagged [phase-estimation]

Refers to the quantum algorithm used to estimate the eigenvalue corresponding to an eigenvector of a unitary operator. (Wikipedia)

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### Why does the “Phase Kickback” mechanism work in the Quantum phase estimation algorithm?

I've probably read the chapter The quantum Fourier transform and its applications from Nielsen and Chuang (10 th anniversary edition) a couple of times before and this took this thing for granted, but ...
984 views

### Quantum algorithm for linear systems of equations (HHL09): Step 1 - Confusion regarding the usage of phase estimation algorithm

I have been trying to get my head around the famous(?) paper Quantum algorithm for linear systems of equations (Harrow, Hassidim & Lloyd, 2009) (more popularly known as the HHL09 algorithm paper) ...
619 views

### Quantum phase estimation and HHL algorithm - knowledge of eigenvalues required?

The quantum phase estimation algorithm (QPE) computes an approximation of the eigenvalue associated to a given eigenvector of a quantum gate $U$. Formally, let $\left|\psi\right>$ be an ...
292 views

### SWAP gate(s) in the $R(\lambda^{-1})$ step of the HHL circuit for $4\times 4$ systems

Context: On the 5th page of the paper Quantum circuit design for solving linear systems of equations (Cao et al, 2012) there's this circuit: Schematic: A brief schematic of what's actually ...
132 views

### Phase estimation algorithm: probability bound of obtaining $m$

Note: Cross-posted on Physics SE. Hi, I'm studying the quantum phase estimation algorithm from this book: M.A. Nielsen, I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge Univ. ...
200 views

### What is the complexity of the quantum phase estimation in Grover's algorithm?

Suppose we are using GA (Grover's algorithm) such that we are given it has 2 or more solutions. The search space is of size $N$. We all know Grover's algorithm has, at worst, a time complexity ...
135 views

### Are the squared absolute values of the eigenvalues of a unitary matrix always 1?

I'm going through the phase estimation algorithm, and wanted to sanity-check my calculations by making sure the state I'd calculated was still normalized. It is, assuming the square of the absolute ...
141 views

### Phase estimation error analysis

This question is about Lemma $7.1.2$ in Kaye, Laflamme, and Mosca's textbook: Let $\omega = \frac{x}{2^n} = 0.x_1x_2\ldots x_n$ be some fixed number. The phase estimation algorithm applied to the ...
128 views

### How to analyze highly entangled quantum circuits?

I came across a quantum circuit very similar to the phase estimation circuit, which is shown below: In the ...
22 views

### Obtaining phases of all basis states

I’m wondering that is there a way to separate phases of basis states from the magnitudes (namely putting phases on some new basis states with the same probability magnitude) if the exact values of ...
130 views

### Cannot replicate results in article on pricing financial derivatives on IBM Q

I am trying to implement a circuit for searching for the largest eigenvalue and respective eigenvector of an operator, i.e. phase estimation, introduced in article Towards Pricing Financial ...
69 views

### Sketch the quantum logic gates correctly and give a proof for the identity

If I denote by $U^c$ the controlled version of the quantum operation $U$ $$U^c=|0\rangle \langle 0|\otimes \mathbb{1}+|1\rangle \langle 1|\otimes U$$ I can first apply $U^c$ and afterward measure ...
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### Eigenstate of unitary operator used for Order-Finding

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen and Chuang", chapter 5.3.1 introduces the concept of solving the Order-Finding Problem. (Eqn 5.36) states ...
108 views

### Quantum Phase Estimation implementation

I tried implementing quantum phase estimation in qiskit, however, I'm not getting the expected results. I choose a controlled $U1$ gate. First of, I implemented inverse QFT operation (basically a ...
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### Method to find $r$ in the case when $r'$ returned by the continued fractions procedure is a factor of $r$

In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the ...
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### Travelling salesman problem on quantum computer

Recently a pre-print of article Efficient quantum algorithm for solving travelling salesman problem: An IBM quantum experience appeared. The authors use a phase estimation as a core for their ...
### Why must we take multiple measurements for different values of $M$?
The circuit for Kitaev phase estimation is given as: By varying $\theta$, we are able to determine $\sin(2 \pi M \phi_k)$ and $\cos (2 \pi M \phi_k)$ from sampling the circuit and calculating the ...
Is it $\theta$ or $\varphi$ as usually depicted on the Bloch sphere? In other words, is it the angle projected on the $xy$-plane or is it the one on a plane that intersects the $z$-axis of the Bloch ...